\newproblem{lay:4_5_21}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 4.5.21}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	The first four Hermite polynomials are $1$, $2t$, $-2+4t^2$ and $-12t+8t^3$. These polynomials arise naturally in the study of certain important differential
	equations in mathematical physics. Show that the first four Hermite polynomials form a basis of $\mathbb{P}_3$.
}{
  % Solution
	Consider the standard basis of $\mathbb{P}_3$:
	\begin{center}
		$E=\{1,t,t^2,t^3\}$
	\end{center}
	In order to know whether the four Hermite polynomials are linearly independent or not we resort to the following augmented matrix whose
	columns are the expression of the Hermite polynomials in the standard basis of $\mathbb{P}_3$
	\begin{center}
		$\left(\begin{array}{rrrr|r} 1 & 0 & -2 & 0 & 0 \\ 0 & 2 & 0 & -12 & 0 \\ 0 & 0 & 4 & 0 & 0 \\ 0 & 0 & 0 & 8 & 0 \end{array}\right) \sim
		 \left(\begin{array}{rrrr|r} 1 & 0 &  0 & 0 & 0 \\ 0 & 1 & 0 &   0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}\right)$
	\end{center}
	So the four Hermite polynomials are linearly independent. Since they are 4 and the dimension of $\mathbb{P}_3$ is also 4, then by Theorem 9.4 of Chapter 5, the
	four Hermite polynomials are a basis of $\mathbb{P}_3$.
}
\useproblem{lay:4_5_21}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
